Chapter 5 Discrete Probability Distributions Statistics for Managers using Microsoft Excel 6th Global Edition Chapter 5 Discrete Probability Distributions Copyright ©2011 Pearson Education

Learning Objectives In this chapter, you learn: The properties of a probability distribution To calculate the expected value and variance of a probability distribution To calculate the covariance and understand its use in finance To calculate probabilities from binomial, hypergeometric, and Poisson distributions How to use the binomial, hypergeometric, and Poisson distributions to solve business problems Copyright ©2011 Pearson Education

Definitions Random Variables A random variable represents a possible numerical value from an uncertain event. Discrete random variables produce outcomes that come from a counting process (e.g. number of classes you are taking). Continuous random variables produce outcomes that come from a measurement (e.g. your annual salary, or your weight). Copyright ©2011 Pearson Education

Definitions Random Variables Ch. 5 Ch. 5 Ch. 5 Discrete Random Variable Discrete Random Variable Discrete Random Variable Continuous Random Variable Continuous Random Variable Continuous Random Variable Ch. 6 Ch. 6 Ch. 6 Copyright ©2011 Pearson Education

Discrete Random Variables Can only assume a countable number of values Examples: Roll a die twice Let X be the number of times 4 occurs (then X could be 0, 1, or 2 times) Toss a coin 5 times. Let X be the number of heads (then X = 0, 1, 2, 3, 4, or 5) Copyright ©2011 Pearson Education

Probability Distribution For A Discrete Random Variable A probability distribution for a discrete random variable is a mutually exclusive listing of all possible numerical outcomes for that variable and a probability of occurrence associated with each outcome. Number of Classes Taken Probability 2 0.20 3 0.40 4 0.24 5 0.16 Copyright ©2011 Pearson Education

Example of a Discrete Random Variable Probability Distribution Experiment: Toss 2 Coins. Let X = # heads. 4 possible outcomes Probability Distribution T T X Value Probability 0 1/4 = 0.25 1 2/4 = 0.50 2 1/4 = 0.25 T H H T 0.50 0.25 Probability H H 0 1 2 X Copyright ©2011 Pearson Education

Discrete Random Variables Expected Value (Measuring Center) Expected Value (or mean) of a discrete random variable (Weighted Average) Example: Toss 2 coins, X = # of heads, compute expected value of X: E(X) = ((0)(0.25) + (1)(0.50) + (2)(0.25)) = 1.0 X P(X) 0 0.25 1 0.50 2 0.25 Copyright ©2011 Pearson Education

Discrete Random Variables Measuring Dispersion Variance of a discrete random variable Standard Deviation of a discrete random variable where: E(X) = Expected value of the discrete random variable X Xi = the ith outcome of X P(Xi) = Probability of the ith occurrence of X Copyright ©2011 Pearson Education

Discrete Random Variables Measuring Dispersion (continued) Example: Toss 2 coins, X = # heads, compute standard deviation (recall E(X) = 1) Possible number of heads = 0, 1, or 2 Copyright ©2011 Pearson Education, Inc. publishing as Prentice Hall Copyright ©2011 Pearson Education

Covariance The covariance measures the strength of the linear relationship between two discrete random variables X and Y. A positive covariance indicates a positive relationship. A negative covariance indicates a negative relationship. Copyright ©2011 Pearson Education

The Covariance Formula where: X = discrete random variable X Xi = the ith outcome of X Y = discrete random variable Y Yi = the ith outcome of Y P(XiYi) = probability of occurrence of the ith outcome of X and the ith outcome of Y Copyright ©2011 Pearson Education

Investment Returns The Mean Consider the return per $1000 for two types of investments. Economic Condition Prob. Investment Passive Fund X Aggressive Fund Y 0.2 Recession - $25 - $200 0.5 Stable Economy + $50 + $60 0.3 Expanding Economy + $100 + $350 Copyright ©2011 Pearson Education

Investment Returns The Mean E(X) = μX = (-25)(.2) +(50)(.5) + (100)(.3) = 50 E(Y) = μY = (-200)(.2) +(60)(.5) + (350)(.3) = 95 Interpretation: Fund X is averaging a $50.00 return and fund Y is averaging a $95.00 return per $1000 invested. Copyright ©2011 Pearson Education

Investment Returns Standard Deviation Interpretation: Even though fund Y has a higher average return, it is subject to much more variability and the probability of loss is higher. Copyright ©2011 Pearson Education

Investment Returns Covariance Interpretation: Since the covariance is large and positive, there is a positive relationship between the two investment funds, meaning that they will likely rise and fall together. Copyright ©2011 Pearson Education

The Sum of Two Random Variables Expected Value of the sum of two random variables: Variance of the sum of two random variables: Standard deviation of the sum of two random variables: Copyright ©2011 Pearson Education

Portfolio Expected Return and Expected Risk Investment portfolios usually contain several different funds (random variables) The expected return and standard deviation of two funds together can now be calculated. Investment Objective: Maximize return (mean) while minimizing risk (standard deviation). Copyright ©2011 Pearson Education

Portfolio Expected Return and Portfolio Risk Portfolio expected return (weighted average return): Portfolio risk (weighted variability) Where w = proportion of portfolio value in asset X (1 - w) = proportion of portfolio value in asset Y Copyright ©2011 Pearson Education

Portfolio Example Investment X: μX = 50 σX = 43.30 Investment Y: μY = 95 σY = 193.21 σXY = 8250 Suppose 40% of the portfolio is in Investment X and 60% is in Investment Y: The portfolio return and portfolio variability are between the values for investments X and Y considered individually Copyright ©2011 Pearson Education

Probability Distributions Ch. 5 Discrete Probability Distributions Continuous Probability Distributions Ch. 6 Binomial Normal Poisson Uniform Hypergeometric Exponential Copyright ©2011 Pearson Education

Binomial Probability Distribution A fixed number of observations, n e.g., 15 tosses of a coin; ten light bulbs taken from a warehouse Each observation is categorized as to whether or not the “event of interest” occurred e.g., head or tail in each toss of a coin; defective or not defective light bulb Since these two categories are mutually exclusive and collectively exhaustive When the probability of the event of interest is represented as π, then the probability of the event of interest not occurring is 1 - π Constant probability for the event of interest occurring (π) for each observation Probability of getting a tail is the same each time we toss the coin Copyright ©2011 Pearson Education

Binomial Probability Distribution (continued) Observations are independent The outcome of one observation does not affect the outcome of the other Two sampling methods deliver independence Infinite population without replacement Finite population with replacement Copyright ©2011 Pearson Education

Possible Applications for the Binomial Distribution A manufacturing plant labels items as either defective or acceptable A firm bidding for contracts will either get a contract or not A marketing research firm receives survey responses of “yes I will buy” or “no I will not” New job applicants either accept the offer or reject it Copyright ©2011 Pearson Education

The Binomial Distribution Counting Techniques Suppose the event of interest is obtaining heads on the toss of a fair coin. You are to toss the coin three times. In how many ways can you get two heads? Possible ways: HHT, HTH, THH, so there are three ways you can getting two heads. This situation is fairly simple. We need to be able to count the number of ways for more complicated situations. Copyright ©2011 Pearson Education

Counting Techniques Rule of Combinations The number of combinations of selecting X objects out of n objects is where: n! =(n)(n - 1)(n - 2) . . . (2)(1) X! = (X)(X - 1)(X - 2) . . . (2)(1) 0! = 1 (by definition) Copyright ©2011 Pearson Education

Counting Techniques Rule of Combinations How many possible 3 scoop combinations could you create at an ice cream parlor if you have 31 flavors to select from? The total choices is n = 31, and we select X = 3. Copyright ©2011 Pearson Education

Binomial Distribution Formula ! x n - x P(X=x |n,π) = π (1-π) x! ( n - x ) ! P(X=x|n,π) = probability of x events of interest in n trials, with the probability of an “event of interest” being π for each trial x = number of “events of interest” in sample, (x = 0, 1, 2, ..., n) n = sample size (number of trials or observations) π = probability of “event of interest” Example: Flip a coin four times, let x = # heads: n = 4 π = 0.5 1 - π = (1 - 0.5) = 0.5 X = 0, 1, 2, 3, 4 Copyright ©2011 Pearson Education

Example: Calculating a Binomial Probability What is the probability of one success in five observations if the probability of an event of interest is 0.1? x = 1, n = 5, and π = 0.1 Copyright ©2011 Pearson Education

The Binomial Distribution Example Suppose the probability of purchasing a defective computer is 0.02. What is the probability of purchasing 2 defective computers in a group of 10? x = 2, n = 10, and π = 0.02 Copyright ©2011 Pearson Education

The Binomial Distribution Shape The shape of the binomial distribution depends on the values of π and n P(X=x|5, 0.1) .6 .4 .2 Here, n = 5 and π = .1 1 2 3 4 5 x P(X=x|5, 0.5) .6 .4 Here, n = 5 and π = .5 .2 1 2 3 4 5 x Copyright ©2011 Pearson Education

The Binomial Distribution Using Binomial Tables (Available On Line) x … π=.20 π=.25 π=.30 π=.35 π=.40 π=.45 π=.50 1 2 3 4 5 6 7 8 9 10 0.1074 0.2684 0.3020 0.2013 0.0881 0.0264 0.0055 0.0008 0.0001 0.0000 0.0563 0.1877 0.2816 0.2503 0.1460 0.0584 0.0162 0.0031 0.0004 0.0282 0.1211 0.2335 0.2668 0.2001 0.1029 0.0368 0.0090 0.0014 0.0135 0.0725 0.1757 0.2522 0.2377 0.1536 0.0689 0.0212 0.0043 0.0005 0.0060 0.0403 0.1209 0.2150 0.2508 0.2007 0.1115 0.0425 0.0106 0.0016 0.0025 0.0207 0.0763 0.1665 0.2384 0.2340 0.1596 0.0746 0.0229 0.0042 0.0003 0.0010 0.0098 0.0439 0.1172 0.2051 0.2461 π=.80 π=.75 π=.70 π=.65 π=.60 π=.55 Examples: n = 10, π = 0.35, x = 3: P(X = 3|10, 0.35) = 0.2522 n = 10, π = 0.75, x = 8: P(X = 8|10, 0.75) = 0.0004 Copyright ©2011 Pearson Education

Binomial Distribution Characteristics Mean Variance and Standard Deviation Where n = sample size π = probability of the event of interest for any trial (1 – π) = probability of no event of interest for any trial Copyright ©2011 Pearson Education

The Binomial Distribution Characteristics Examples P(X=x|5, 0.1) .6 .4 .2 1 2 3 4 5 x P(X=x|5, 0.5) .6 .4 .2 1 2 3 4 5 x Copyright ©2011 Pearson Education

Using Excel For The Binomial Distribution Copyright ©2011 Pearson Education

The Poisson Distribution Definitions You use the Poisson distribution when you are interested in the number of times an event occurs in a given area of opportunity. An area of opportunity is a continuous unit or interval of time, volume, or such area in which more than one occurrence of an event can occur. The number of scratches in a car’s paint The number of mosquito bites on a person The number of computer crashes in a day Copyright ©2011 Pearson Education

The Poisson Distribution Apply the Poisson Distribution when: You wish to count the number of times an event occurs in a given area of opportunity The probability that an event occurs in one area of opportunity is the same for all areas of opportunity The number of events that occur in one area of opportunity is independent of the number of events that occur in the other areas of opportunity The probability that two or more events occur in an area of opportunity approaches zero as the area of opportunity becomes smaller The average number of events per unit is  (lambda) Copyright ©2011 Pearson Education

Poisson Distribution Formula where: x = number of events in an area of opportunity  = expected number of events e = base of the natural logarithm system (2.71828...) Copyright ©2011 Pearson Education

Poisson Distribution Characteristics Mean Variance and Standard Deviation where  = expected number of events Copyright ©2011 Pearson Education

Using Poisson Tables (Available On Line) X  0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 1 2 3 4 5 6 7 0.9048 0.0905 0.0045 0.0002 0.0000 0.8187 0.1637 0.0164 0.0011 0.0001 0.7408 0.2222 0.0333 0.0033 0.0003 0.6703 0.2681 0.0536 0.0072 0.0007 0.6065 0.3033 0.0758 0.0126 0.0016 0.5488 0.3293 0.0988 0.0198 0.0030 0.0004 0.4966 0.3476 0.1217 0.0284 0.0050 0.4493 0.3595 0.1438 0.0383 0.0077 0.0012 0.4066 0.3659 0.1647 0.0494 0.0111 0.0020 Example: Find P(X = 2 |  = 0.50) Copyright ©2011 Pearson Education

Using Excel For The Poisson Distribution Copyright ©2011 Pearson Education

Graph of Poisson Probabilities Graphically:  = 0.50 X  = 0.50 1 2 3 4 5 6 7 0.6065 0.3033 0.0758 0.0126 0.0016 0.0002 0.0000 P(X = 2 | =0.50) = 0.0758 Copyright ©2011 Pearson Education

Poisson Distribution Shape The shape of the Poisson Distribution depends on the parameter  :  = 3.00  = 0.50 Copyright ©2011 Pearson Education

The Hypergeometric Distribution The binomial distribution is applicable when selecting from a finite population with replacement or from an infinite population without replacement. The hypergeometric distribution is applicable when selecting from a finite population without replacement. Copyright ©2011 Pearson Education

The Hypergeometric Distribution “n” trials in a sample taken from a finite population of size N Sample taken without replacement Outcomes of trials are dependent Concerned with finding the probability of “X” items of interest in the sample where there are “A” items of interest in the population Copyright ©2011 Pearson Education

Hypergeometric Distribution Formula Where N = population size A = number of items of interest in the population N – A = number of events not of interest in the population n = sample size x = number of items of interest in the sample n – x = number of events not of interest in the sample Copyright ©2011 Pearson Education

Properties of the Hypergeometric Distribution The mean of the hypergeometric distribution is The standard deviation is Where is called the “Finite Population Correction Factor” from sampling without replacement from a finite population Copyright ©2011 Pearson Education

Using the Hypergeometric Distribution Example: 3 different computers are checked out from 10 in the department. 4 of the 10 computers have illegal software loaded. What is the probability that 2 of the 3 selected computers have illegal software loaded? N = 10 n = 3 A = 4 x = 2 The probability that 2 of the 3 selected computers have illegal software loaded is 0.30, or 30%. Copyright ©2011 Pearson Education

Using Excel for the Hypergeometric Distribution Copyright ©2011 Pearson Education

Chapter Summary Addressed the probability distribution of a discrete random variable Defined covariance and discussed its application in finance Discussed the Binomial distribution Discussed the Poisson distribution Discussed the Hypergeometric distribution Copyright ©2011 Pearson Education

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